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Interval Computations in Metrology

Vladik Kreinovich1, Konstantin Semenov2, and Gennady N. Solopchenko2

1Department of Computer Science, University of Texas at El Paso

El Paso, TX 79968, USA, vladik@utep.edu

2Peter the Great St. Petersburg Polytechnic University

29 Polytechnicheskaya str., St. Petersburg, 195251, Russia semenov.k.k@iit.icc.spbstu.ru, g.n.solopchenko@iit.icc.spbstu.ru

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1. Need for Data Processing

• In many engineering situations, we need to make deci-

sions.

• Some of these decisions are made by humans, some by

automatic control systems.

• The decisions y are based on the valued of the relevant

quantities x1, . . . , xn: y = f(x1, . . . , xn).

• Ideally, the values xi should come from measurement.
• However, in many cases, we also need to use expert

estimates.

• This is typical, e.g., in inverse problems, which are, in

general, ill-defined.

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2. Need for Expert Estimates

• For example, we may be interested in the value x(t),

but sensors only measure averages xav(t) = t+ε

t−ε

K(t − t′) · x(t′) dt and

• K(τ) dτ = 1.
• To make these problems well-defined, we need to add

prior information – which comes from experts.

• For example, in measuring x(t), the experts can give

us the upper bound M on the rate of change | ˙ x(t)|.

• In this case, |x(t) − xav(t)| ≤ M · ε.
• Both measurement results and expert estimates come

with uncertainty.

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3. Need to Take Uncertainty into Account

• Measurements are never absolutely accurate.
• The measurement result

x is, in general, different from the actual value x of the corresponding quantity.

• Ideally, we should know the probability distribution for

the measurement error ∆x

def

= x − x.

• However, in most practical cases, all we know is the

upper bound ∆ on the measurement error: |∆x| ≤ ∆.

• In this case, once we have a measurement result

x, all we know about the actual value x is that x ∈ [ x − ∆, x + ∆].

• Expert estimates are also imprecise.

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4. Gauging Expert Uncertainty

• Ideally, we should view each expert as a measuring

instrument: – we compare expert estimates and measurement re- sults, and – we get a probability distribution for the estimation error ∆x = x − x.

• In practice, we rarely have enough samples to make

statistically meaningful estimates.

• A reasonable way to describe expert uncertainty is to

ask the expert to estimate, – for each possible value x ≈ x, – to what extent x is possible.

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5. Gauging Expert Uncertainty (cont-d)

• For example, we can ask the expert to mark her cer-

tainty by a mark m on a scale from 0 to s.

• Then we take m/s as the degree.
• The function µ(x) assigning degree to a value x is

known as a fuzzy set.

• If for each variable xi, we only know that xi ∈ xi =

[xi, xi], then we know that y = f(x1, . . . , xn) ∈ y = f(x1, . . . , xn)

def

= {f(x1, . . . , xn) : xi ∈ xi}.

• Computing such a range y is one of the main problems
• f interval computations.

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6. Processing Expert Uncertainty

• For expert estimates, it is reasonable to consider:

– for every α ∈ [0, 1], – the set xi(α) = {xi : µi(xi) ≥ α} of sufficiently possible values.

• Then, for every α, we compute the range

y(α) = f(x1(α), . . . , xn(α)).

• This can also be done by interval computation tech-

niques.

• Additional problems:

– sometimes, the dependence y = f(x1, . . . , xn) is not known exactly; – even when we know the exact dependence, we can

• ften only compute f(x1, . . . , xn) approximately.

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7. Processing Expert Uncertainty (cont-d)

• The approximate character of computing f(x1, . . . , xn)

is caused by: – rounding errors for arithmetic operations, – inevitably imprecise formulas for non-arithmetic el- ementary functions such as exp(x) etc.

• One of the main objectives of metrology is:

– to provide guaranteed information about the actual values of the quantities of interest – based on measurement results and expert esti- mates.

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8. Interval and Fuzzy Computations in Metrol-

• gy: A Brief History
• 1960s: IFIP (led by Wilkinson) proposes:

– accompanying each data processing software – with bounds (interval) estimate of the result’s in- accuracy.

• 1960s: Moore et al.

proposed general interval tech- niques for such estimates.

• 1970s: software packages with guaranteed bounds (e.g.,

Linpack).

• 1965: fuzzy sets introduced by Zadeh.
• 1980s: L. K. Reznik combined expert estimates with

measurement intervals in practical problems.

• 1985: first standard for metrological support of data

processing.

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9. Interval and Fuzzy Computations in Metrol-

• gy: A Brief History (cont-d)
• 1985: systematic way of providing such support de-

scribed in a special issue of Measuring Techniques.

• 1990s: further theoretical development and algorithms

design.

• 2000s–2010s: metrological proposals for taking interval

and fuzzy uncertainty into account.

• What we would like: to incorporate interval and fuzzy

techniques in metrological practice.

• What is needed for this:

– add interval and fuzzy computations to the existing metrological standards, – make the corresponding algorithms as simple as possible and as clear to engineers as possible.

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10. Case Study: Heat Meter

• In many practical situations, we need to know how

much heat or cooling was generated or consumed.

• For example, in nuclear power stations:

– water or gas is heated by a reactor, – the steam is moved to a turbine that generates elec- tricity, – when the steam rotates the turbine, it loses energy and cools down.

• Similarly, in heating and air conditioning systems:

– hot water is circulated, heating a building; – cold air is circulated, cooling the building; – in dry areas, water is used to cool the buildings.

• In all these cases, it is desirable to measure the amount
• f heat.

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11. Case Study: Heat Meter (cont-d)

• The amount is difficult to measure directly.
• So, heat meters measure flow rate, pressure, in- and
• ut-temperatures and compute the heat flow as

flow rateout · t◦

• ut − flow ratein · t◦

in.

• Existing standards only take into account uncertainty

in temperature sensors.

• Thus, the existing method underestimate measurement

error.

• There is also uncertainty is measuring flow rate.
• Some of this uncertainty comes from inhomogeneity,

which needs expert estimates.

• We (K.S. and G.S.) took this into account and got es-

timates consistent with more accurate measurements.

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12. Acknowledgments This work was supported in part:

• by the National Science Foundation grants:
• HRD-0734825 and HRD-1242122

(Cyber-ShARE Center of Excellence) and

• DUE-0926721, and
• by an award from Prudential Foundation.